I solved the missing paper/take home exam problem years ago. When the students turn in their papers/exams, I have them sign in on a folder as they place them into it. Then we all know who turned in their papers/exams since there is or is not a signature. This also helps me keep track of whose work is missing.
I like this idea, which came from the Chronicle forums.
I will have to think about whether or not I should institute it.